+  = 9216 = 96^2.       (1)


Therefore,   =  - .    (2)


Hence,   = (y+96)*(y-96).      (3)


Thus,   is the product of integers y+96 and y-96.


From it, we conclude that  y+96 and y-96 are degrees of 2.

So, we write

    y + 96 = 

    y - 96 = 


with integer non-negative n and m, and we understand that m < n.  
Subtracting the lower equation from the upper one, we get

     -  = 192,

      = 192 = 64*3 = .   (4)


From it, we conclude that  m = 6, n-m = 2;  hence n-6 = 2,  n = 8.


Thus    y+96 =  =  = 256;  hence  y = 256-96 = 160.

CHECK:  y-96 =  =  = 64;  hence  y = 64+96 =160.   <<<---===  Check says OK.


Now  from (3)  

     = (y+96)*(y-96) = (160+96)*(160-96) = 16384 = ,


Hence,  x = 14.


ANSWER.  This problem has two solutions  (x,y) = (14,160)  and  (x,y) = (14,-160).


From where the second, negative value of y came ?


Since right side of equation (1) is y^2, it is clear that with positive solution y= 160,
negative solution y= -160 works, too.


Where we missed it in our reasoning ? - Because in (4), we could take NEGATIVE factors    and  -3 
into consideration.  It would lead us to the negative value of y.

But since we just caught this second solution, we shouldn't worry anymore.